我們研究具有 PT 對稱勢的非線性薛丁格方程式，該方程式在材料科學、物理學和量子力學乃至於研究化學中，皆為一項重要的方程式，PT-對稱性分為宇稱對稱性和時間反演對稱性，宇稱對稱性是指所有坐標的符號反轉，時間反演對稱性是指時間變量符號的反轉。本論文透過運用物理信息神經網絡深度學習來尋找此系統在兩種不同 PT 對稱勢下的波函數，並將這些結果與克蘭克－尼科爾森方法所得出的結果進行比較，在程式語言上，我們選用 Python 以及Tensorflow 框架的編寫與訓練，這是一種成熟的大型深度學習軟體庫。優化器選擇了最廣泛使用的擬牛頓法－ BFGS 法應用此神經網路上，方法具有二階優化器的收斂速度優勢，並且計算複雜度低於牛頓法。我們也研究了對於神經網路的不同超參數，例如層數和神經元的數量調整，對物理信息神經網絡深度學習具有 PT 對稱勢的非線性薛丁格方程式的影響。

We study the nonlinear Schrödinger equation (NLSE) with the PT-symmetric potential, which is an classical equation in physics and quantum mechanics. PT-symmetry is divided into parity symmetry and time-reversal symmetry. Parity symmetry refers to the change of the sign of all coordinates, and time-reversal symmetry refers to the change of the sign of the time variable.
We apply physics-informed neural networks (PINNs) to find the wave functions of this system under two different PT-symmetric potentials, and the results are compared with those of Crank-Nicolson method. Here we use Python and the Tensorflow suite to implement the code of the deep neural networks. BFGS -one of the most widely used quasi-Newton methods, was chosen as the optimizer in PINNs. We also investigate the influence of the layers and neurons on the learning ability of the PINN method in solving the NLSE with the PT-symmetric potential. The results indicate that the proposed PINNs are effective to solve the wave functions of the NLSE under the PT-symmetric potentials.

Abstract. . . . . . . . . .i
1 Introduction. . . . . . . . . .1
2 Neural Network. . . . . . . . . .3
2.1 Model . . . . . . . . . .3
2.1.1 Physics-informed neural networks. . . . . . . . . .3
2.1.2 The solutions of the NLSE with PT-symmetric harmonic potential. . . . . . . . . .4
2.1.3 The solutions of the NLSE with PT-symmetric Scarf-II potential. . . . . . . . . .7
2.2 Optimizer. . . . . . . . . .8
2.2.1 Newton’s method. . . . . . . . . .9
2.2.2 Quasi-Newton method (BFGS). . . . . . . . . .10
2.3 Initialization. . . . . . . . . .12
2.3.1 Xavier initialization. . . . . . . . . .13
2.4 Finite-difference Method. . . . . . . . . .14
2.4.1 Forward difference. . . . . . . . . .15
2.4.2 Backward difference. . . . . . . . . .16
2.4.3 Crank-Nicolson Method. . . . . . . . . .17
3 Results. . . . . . . . . .19
3.1 PT-symmetric harmonic potential. . . . . . . . . .19
3.1.1 The focusing case with initial conditions from exact soliton. . . . . . . . . .19
3.1.2 The defocusing case with initial conditions from exact soliton. . . . . . . . . .20
3.1.3 The periodic initial condition case. . . . . . . . . .21
3.2 PT-symmetric Scarf-II potential. . . . . . . . . .23
3.2.1 The focusing case with initial conditions from exact soliton. . . . . . . . . .23
3.2.2 The defocusing case with initial conditions from exact soliton. . . . . . . . . .24
3.3 Comparison. . . . . . . . . .25
3.3.1 Comparing the accuracy of PINNs and Crank-Nicolson. . . . . . . . . .25
3.3.2 Comparisons of the influence of layers and neurons on the learning ability of this PINN. . . . . . . . . .26
4 Conclusion. . . . . . . . . .28
Reference. . . . . . . . . .29

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